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Miscellanea

Recreational mathematics, Undergraduate Research, Publications for Teachers and the General Public

Among the mathematical researchers at the University of Luxembourg one should count the many students at all levels who carry out original projects for their Bachelor/Master thesis or within the framework of the Experimental Mathematics Lab. The related publications are below, together with books and articles for teachers or the general public. To get information on the many outreach activities of the department of mathematics, please have a look at the Outreach page. 

Graph relating generic curves and homotopy moves; by Charlène Roncalli (Master student) 

 

BOOKS

  • Christophe Ley (Professor), Science Meets Sports: When Statistics Are More Than Numbers, co-editor : Yves Dominicy, Cambridge Scholars Publishing, 2020, ISBN-10: 1527558568, ISBN-13: 978-1527558564.

This book is a collection of chapters covering different sports and statistical methods put to good use in the analysis of these sports. The book is designed to be of interest for and readable by both academic and sports audiences, as special new insights into the individual sports are provided through basic and advanced statistics. 

  • Antonella Perucca (Professor) and Lisa Vagnozzi (Children’s Author), Douze histoires fantastiques: Énigmes et aventures pour mathématiciens en herbe, 2022, amazon print-on-demand, ISBN-13 979-8818508863

This is a collection of 12 stories based on mathematical riddles. The solution is given progressively, as the characters solve the enigmas as a joint effort. For anyone aged 10 years or more.

ARTICLES IN JOURNALS AND PROCEEDINGS 

  • Miguel Acosta (Postdoc) and Jean-Marc Schlenker (Professor), A Hyperbolic Proof of Pascal’s Theorem, The Mathematical Intelligencer, 2021, vol. 43, issue 2, No 23, 130-133. 

Blaise Pascal discovered a remarkable theorem in projective geometry, characterizing hexagons inscribed in ellipses by the alignment of the intersection points of opposite edges. The proof given by Pascal, as well as other proofs discovered later, was quite complex and intricate. We realized that a simple proof of Pascal’s theorem can be given using elementary properties of the hyperbolic plane. 

  • Rich Andrusiak (Professor) and Antonella Perucca (Professor), Parallel Computing and the Value of Teamwork, to appear in Mathematics Teacher: Learning and Teaching Pre-K-12, 2022.  

This article contains mathematical exercises for all grades that are an introduction to parallel computing. The exercises investigate the possibility of splitting calculations (even mental calculations!) within a team.  

  • Tom Fiore (Professor), Alexander Lang (Bachelor), and Antonella Perucca (Professor), Tactile tools for teaching: an implementation of Knuth’s algorithm for mastering Mastermind, College Journal of Mathematics, vol. 49, no. 4, September 2018, 278–286. 

Donald E. Knuth found a strategy to always win a Master Mind game in at most 5 moves. If you want to see how it works, then have a look at this article or the related material available on Antonella Perucca’s webpage (https://antonellaperucca.net). 

  • Christophe Ley (Professor), Maths et sports - quand les statistiques sont plus que des nombres, Maths Express, 87–92 (2021). 

This article shows how mathematics, and in particular data science/statistics, are omnipresent in the world of sports nowadays. Some concrete applications from soccer are described.  

  • Antonella Perucca (Professor), Multimengen für die Arithmetik, Proceedings of the GDM Tagung in Regensburg 2019, Beiträge zum Mathematikunterricht 2019, 613–616, http://hdl.handle.net/2003/38628. 

Multisets are easy generalizations of sets that are very useful for every-day purposes because they allow repetitions of the elements. In school, working with the multiset of prime factors of a number provides an interesting point of view on arithmetic.        

  • Antonella Perucca (Professor), The Chinese Remainder Clock, College Journal of Mathematics, vol. 48, no. 2, March 2017, 82–89. 

The Chinese Remainder Clock, invented by the author, is a tool to illustrate the Chinese Remainder Theorem. The article also contains a statement and proof of this fundamental mathematical result without congruences. See also the related website by IMAGINARY (https://imaginary.github.io/chinese-remainder-clock/index.html). 

  • Antonella Perucca (Professor) and Deborah Stranen (Master), Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom Exercise, Convergence: Where Mathematics, History, and Teaching Interact, Mathematical Association of America, vol. 17 (2020), posted 10/26/2020, https://www. maa.org/press/periodicals/convergence/whats-in-convergence-contents-of-volume-17-2020. 

The celebrated Plimpton 322 contains a list of numbers written in base 60 that are related to Pythagorean triples. To facilitate the mathematical understanding of the Babylonian tablet we construct its analogue in base 10 with different numbers. 

  • Antonella Perucca (Professor), Joe Reguengo da Sousa (Bachelor), and Sebastiano Tronto (PhD), Arithmetic billiards, Recreational Mathematics Magazine, vol. 9, 2022, 43–54. 

Arithmetic billiards show a nice interplay between geometry and arithmetic. We generalize the classical results by allowing any starting point for the billiard ball (which is usually shot from a corner).  

  • Antonella Perucca (Professor) and Milko Todorovic (Master), Veranschaulichungen des Induktionsprinzips, Proceedings of the online GDM (Gesellschaft für die Didaktik der Mathematik) Meeting 2020, Beiträge zum Mathematikunterricht 2020, Hans-Stefan Siller, Wolfgang Weigel, and Jan Franz Wörler (Editors), WTM-Verlag Münster, 705–708. 

The domino effect visualizes the principle of mathematical induction. We investigate variants of the mathematical induction and the corresponding variants of the domino effect that are needed to illustrate them.

  • And Foyen (Master student) and Antonella Perucca (Professor), Der 50cm lange Gliedermaßstab, Proceedings of the online GDM (Gesellschaft für die Didaktik der Mathematik) Meeting 2022, Beiträge zum Mathematikunterricht 2022, in press.

This article presents a didactical project on the foldable half-meter, that can be used in particular to explore shapes and could be considered a Montessori tool, see https://math.uni.lu/meter/.

  • Antonella Perucca (Professor) and Emiliano Torti (Postdoc), Congruence theorems for convex polygons involving sides, angles and diagonals, International Journal of Geometry, Vol. 12 (2023), No. 1, 83 - 92.

This article contains new results for general convex n-gons, in particular whether they are determined up to congruence when the lengths of sufficiently many diagonals are known.

  • Francesco Grotto (Postdoc), Antonella Perucca (Professor), and Tatjana Van Steenbergen Bergeron (High-school pupil), Rubik’s Snakes on a plane, College Mathematics Journal, in press.

This article contains mathematical results on the planar configurations of the mathematical toy called Rubik's Snake.

  • Antonella Perucca (Professor), Congruence Theorems in the past, present, and future, For the Learning of Mathematics, in press. 

This article is meant to explain the didactical potential of the congruence theorems to explore mathematics and even to prove new results.

FURTHER PUBLICATIONS 

This article contains insights on managing and maintaining the Stacks project and other large-scale mathematical websites devoted to classification results. 

  • Christophe Ley (Professor), Pierre Mangers (Consultant), and Ivan Nourdin (Professor): Les mathématiques, un grand défi de cette pandémie, Tageblatt, June 2020. 

This article describes how far mathematics plays a role in the COVID-19 pandemics, and explains the basic notions such as exponential growth and difference mean/median, among others.

This article explains the classical results on arithmetic billiards, which for example allow to represent in a geometrical way the greatest common divisor of two positive integers. 

A clock dial is supposed to contain the numbers from 1 to 12 but in fact there are many ways of writing these numbers. For example, it is possible to express them with formulas that only involve the digit 7. 

We suggest treating all minorities as a whole and encourage to pay attention to the special needs of each for a truly inclusive atmosphere. 

  • Antonella Perucca (Professor), The Easy Multiplication Tables, Research Poster for the online conference GDM Tagung 2021, Gesellschaft für die Didaktik der Mathematik. 

For learning by heart the multiplication tables one should exploit commutativity (for example, if you know 6x7, then you already know 7x6). We thus suggest a visualization of the multiplication tables containing only 45 operations. 

Test your logic skills to solve a riddle where you can ask Yes/No questions to three Oracles (one telling the truth, one lying, and one randomly deciding whether to lie or tell the truth). Note that the Oracles will answer in their own language, which you don’t understand. 

To illustrate any recurrence sequence, you can draw its entries on a line and write the recurrence relation on a sliding mask with holes. If you place the sliding mask anywhere on the line, then you see a relation that holds true as the numbers in it are correct. 

  • Antonella Perucca (Professor) and Engjëll Begalla (Master), Luc van den Broek (Teacher), Deborah Stranen (Master), Uitwiskeling: Het ABCD van de koordenvierhoeken, vol. 36 no. 4, 7–10. Driehoeken met onderling 5 gelijke zijden en hoeken, vol. 34, no. 3, 8–10. Veelvouden van 3 graden, vol. 35, no. 2, 7–10. Het kunstgalerijprobleem, vol. 35 no. 3, 2–5. Multiverzameling, vol. 35, no. 4, 3–5. De zeven bruggen van Koningsbergen, vol. 36 no. 1, 8–10. De 15-puzzel, vol. 36 no. 2, 8–9. Vier punten, twee afstanden, vol. 36 no. 3, 13–15. Elk natuurlijk getal is het begin van een macht van twee, vol. 37 no. 2, 23–25. Trapgetallen, vol. 37 no. 3, 18–20.  

The above articles (translated into Dutch) are popularization articles on various topics published in the magazine Uitwiskeling.

  • Antonella Perucca (Professor), Multiplying mathematical teachers, GDM (Gesellschaft für die Didaktik der Mathematik) Mitteilungen, in press.

This communication presents some concrete ideas to face the lack of mathematical teachers.

ONLINE TOOLS

Pieter Belmans (Professor) Fanography https://www.fanography.info/ Le superficie algebriche https://superficie.info/ grassmannian.info https://www.grassmannian.info/ Hyperkaehler.info https://www.hyperkaehler.info/

The above websites are, respectively:

  • A tool to visually study the geography of Fano 3-folds.
  • A tool to visually study the geography of (minimal) complex algebraic smooth surfaces by Enriques.
  • A periodic table of (generalised) Grassmannians.
  • A description of the geography of compact irreducible holomorphic symplectic (or hyperkähler) varieties.

Daniel Ramos (Programmer) and Antonella Perucca (Professor), The Chinese Remainder Clock https://imaginary.github.io/chinese-remainder-clock/index.html

This page contains a working and interactive Chinese Remainder Clock and some explanations of the mathematics behind it. 

VIDEOS 

Why is it important to know statistics? Conference "Science and Politics Interface at the Parliament", organized by the Cellule Scientifique of the Luxembourgish Parliament.

The mathematics behind COVID 19 explained (in Dutch) for a lecture at Ghent University.  

What are Artificial Intelligence and Machine Learning, and how can their potential be fully unleashed in sports medicine? Conference Sportmedica 2021 Luxembourg. 

The Rubik’s Cube world record holder Sebastiano Tronto explains some mathematics related to the Rubik’s Cube and amazes the audience with his extraordinary skills. 

An outreach talk at the Luxembourg Science Center over AI and Data Science.

Podcast interview with SciLux, recorded at RTL, over Statistics, Data Science and Sports.

CONTACT

Professor Antonella Perucca, antonella.perucca@uni.lu